3.1825 \(\int \frac{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^4} \, dx\)

Optimal. Leaf size=35 \[ \frac{(a e+c d x)^2}{2 (d+e x)^2 \left (c d^2-a e^2\right )} \]

[Out]

(a*e + c*d*x)^2/(2*(c*d^2 - a*e^2)*(d + e*x)^2)

_______________________________________________________________________________________

Rubi [A]  time = 0.0412381, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ \frac{(a e+c d x)^2}{2 (d+e x)^2 \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^4,x]

[Out]

(a*e + c*d*x)^2/(2*(c*d^2 - a*e^2)*(d + e*x)^2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.1162, size = 29, normalized size = 0.83 \[ - \frac{\left (a e + c d x\right )^{2}}{2 \left (d + e x\right )^{2} \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)/(e*x+d)**4,x)

[Out]

-(a*e + c*d*x)**2/(2*(d + e*x)**2*(a*e**2 - c*d**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0205752, size = 29, normalized size = 0.83 \[ -\frac{a e^2+c d (d+2 e x)}{2 e^2 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^4,x]

[Out]

-(a*e^2 + c*d*(d + 2*e*x))/(2*e^2*(d + e*x)^2)

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 40, normalized size = 1.1 \[ -{\frac{a{e}^{2}-c{d}^{2}}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{cd}{{e}^{2} \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)/(e*x+d)^4,x)

[Out]

-1/2*(a*e^2-c*d^2)/e^2/(e*x+d)^2-c*d/e^2/(e*x+d)

_______________________________________________________________________________________

Maxima [A]  time = 0.723461, size = 58, normalized size = 1.66 \[ -\frac{2 \, c d e x + c d^{2} + a e^{2}}{2 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^4,x, algorithm="maxima")

[Out]

-1/2*(2*c*d*e*x + c*d^2 + a*e^2)/(e^4*x^2 + 2*d*e^3*x + d^2*e^2)

_______________________________________________________________________________________

Fricas [A]  time = 0.201392, size = 58, normalized size = 1.66 \[ -\frac{2 \, c d e x + c d^{2} + a e^{2}}{2 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^4,x, algorithm="fricas")

[Out]

-1/2*(2*c*d*e*x + c*d^2 + a*e^2)/(e^4*x^2 + 2*d*e^3*x + d^2*e^2)

_______________________________________________________________________________________

Sympy [A]  time = 1.77239, size = 44, normalized size = 1.26 \[ - \frac{a e^{2} + c d^{2} + 2 c d e x}{2 d^{2} e^{2} + 4 d e^{3} x + 2 e^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)/(e*x+d)**4,x)

[Out]

-(a*e**2 + c*d**2 + 2*c*d*e*x)/(2*d**2*e**2 + 4*d*e**3*x + 2*e**4*x**2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.206132, size = 62, normalized size = 1.77 \[ -\frac{{\left (2 \, c d x^{2} e^{2} + 3 \, c d^{2} x e + c d^{3} + a x e^{3} + a d e^{2}\right )} e^{\left (-2\right )}}{2 \,{\left (x e + d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(e*x + d)^4,x, algorithm="giac")

[Out]

-1/2*(2*c*d*x^2*e^2 + 3*c*d^2*x*e + c*d^3 + a*x*e^3 + a*d*e^2)*e^(-2)/(x*e + d)^
3